Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities

In this paper we study manifolds, $X_{\mathrm{\Sigma }}$, with fibred singularities, more specifically, a relevant space ${\mathcal{R}}^{\mathrm{p}\mathrm{s}\mathrm{c}}(X_{\mathrm{\Sigma }})$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space ${\mathcal{R}}^{\mathrm{p}\mathrm{s}\mathrm{c}}(X_{\mathrm{\Sigma }})$ is homotopy invariant under certain surgeries on $X_{\mathrm{\Sigma }}$.